The present invention is directed, in general, to systems and methods for performing circuit analysis and, more specifically, to a system and method for analyzing a circuit that incorporates a either forced or unforced oscillators.
Oscillatory behavior is ubiquitous in nature and can be found in a variety of electrical, mechanical, gravitational and biological systems. In electronic oscillators, salient examples include voltage-controlled oscillators (VCOs), phase-locked loops (PLLs), frequency dividers and sigma delta modulators. In the presence of external forcing, these systems can exhibit complex dynamics, such as frequency modulation, entrainment or mode locking, period multiplication and even chaos behavior. Despite their universality, our understanding of such systems is far from complete, and it is difficult to predict the response of a general autonomous system in a satisfactory and reliable manner.
Historically, many previous oscillator investigations have been viewed from a practical design perspective wherein their analyses have typically applied purely linear concepts to obtain simple design formulas. However, linear models are not qualitatively adequate for practical oscillators, since nonlinearity is essential for their orbital stability. Nonlinear analyses have largely concentrated on polynomially-perturbed linear oscillators with sophisticated studies focusing typically on mode locking and transitions to chaos. Relatively little attention has been paid to phenomena like FM-quasiperiodicity, even though they are of great importance in communication applications.
A current analytical technique is the multiple-variable expansion procedure, which is useful for simple harmonic oscillators that do not have external forcing and exhibit small nonlinear perturbations. The dependence on the strength of the nonlinearity is typically different in different parts of the solution, causing multiple time variables to be introduced to obtain a tractable perturbation theory. Unfortunately, this method is intrinsically a perturbation approach, wherein even convergence of the solution series is not guaranteed.
For real oscillators and more complex systems, numerical simulation has been the predominant means of predicting detailed responses. However, simulation of oscillators presents unique difficulties that are absent in non-autonomous systems. A fundamental problem is the intrinsic phase instability of oscillators, which is manifested as the absence of a time reference. In this case, numerical errors tend to grow and the phase error often increases without bound in the course of the numerical solution. For unforced oscillators in periodic steady state, boundary-value methods such as shooting and harmonic balance can be used to obtain both the time period and the steady-state solution. Neither shooting nor harmonic balance can be applied, however, to forced oscillators with FM-quasiperiodic responses, as they require an impractically large number of time-steps or variables. In practice, the separation of the time scales is often reduced artificially to make the problem tractable. Such ad hoc approaches can lead to qualitatively misleading results.
Accordingly, what is needed in the art is a way to transform a frequency modulated quasiperiodic waveform, such as that from an oscillator, to enhance and simplify its analysis.
To address the above-discussed deficiencies of the prior art, the present invention provides a system for, and method of, analyzing a forced or unforced oscillator. In one embodiment the system includes: (1) a transformation circuit that transforms a frequency-modulated waveform representing an output of the oscillator into a function based on at least two time scales and (2) a numeric analyzer, associated with the transformation circuit, that warps at least one of the at least two time scales and thereafter numerically analyzes the function to determine a frequency thereof.
The present invention therefore introduces the broad concept of causing at least one of the multiple time scales to vary (warp) Warping of the time scale causes the oscillator output waveform to become quasiperiodic, greatly reducing the numeric analysis required to determine frequencies and substantially eliminating the error that results when approximations (that otherwise would be required were the time scale to be fixed) are rendered unnecessary. Warping as used here means to stretch or squeeze the time axis by different amounts at different times in order to make the density of the oscillator output waveform undulations uniform.
In one embodiment of the present invention, a first of the at least two time scales is at least three times faster than a second of the at least two time scales. In an embodiment to be illustrated and described, the time scales are separated greatly, perhaps on the order of 50 times.
In one embodiment of the present invention, the function is a multirate partial differential equation (MPDE). Those skilled in the pertinent art are familiar with MPDEs and their solutions.
In one embodiment of the present invention, the numeric analyzer analyzes the function by performing a Fourier transform thereon. Those skilled in the pertinent art are familiar with Fourier transforms and their use in analyzing oscillators in a frequency domain. Of course, the present invention also encompasses time-domain analysis.
In one embodiment of the present invention, the frequency-modulated waveform comprises a plurality of voltage samples. Alternatively, the waveform may be represented by an equation.
In one embodiment of the present invention, the oscillator is a forced oscillator. Alternatively, the oscillator may be unforced. The present invention provides a general method of solution for both forced and unforced oscillators.
In one embodiment of the present invention, the oscillator is a voltage controlled oscillator. Of course, those skilled in the pertinent art will perceive a wide variety of electrical, mechanical, gravitational and biological applications for the system and method of the present invention.
The foregoing has outlined, rather broadly, preferred and alternative features of the present invention so that those skilled in the art may better understand the detailed description of the invention that follows. Additional features of the invention will be described hereinafter that form the subject of the claims of the invention. Those skilled in the art should appreciate that they can readily use the disclosed conception and specific embodiment as a basis for designing or modifying other structures for carrying out the same purposes of the present invention. Those skilled in the art should also realize that such equivalent constructions do not depart from the spirit and scope of the invention in its broadest form.